34 problems

Let $b$ be the base and let $D_1,D_2\subseteq\{0,1,\ldots,b-1\}$ be digit sets, with $\#D_1$ and $\#D_2$ denoting their cardinalities. A pair $(D_1,D_2)$ is called degenerate or Ke…

Super-exponential condensation conjecture. Any analytic IFS which has super-exponential condensation but no exact overlaps must be sub-conjugated to a self-similar IFS.

Logarithmic commensurability conjecture. The contraction ratios are logarithmically commensurable, meaning

Exact overlap conjecture. If

Let $1/2<\rho<1$ be a scaling factor, and let $\{\alpha^i\}_{i\in I}$ be the reduced banned words, enumerated in nondecreasing order of length, with $I\subseteq\mathbb{N}$. For eac…

Let $\Phi = \{x\mapsto r_i x + a_i\}_{i\in {\mathcal I}}$ be a self-similar iterated function system on $\mathbb R$, with $|r_i|\in (0,1)$, and let $\Lambda_\Phi$ be its attractor.…

Let $(A,L)$ satisfy the requirements of the strictly contractive dual iterated function system, let $\phi_0\in\mathcal{S}_{\mathrm{c}}(A)$, and let $\phi_\pi$ be the self-similar s…

Let $(A,L)$ satisfy the requirements of the strictly contractive dual iterated function system, and let $\phi_0\in\mathcal{S}_{\mathrm{c}}(A)$. Purity and compact-support conjectur…

Let $({\mathbb X},d)$ be a metric space, and call $[x,y]:=\{z\in{\mathbb X}:d(x,z)+d(z,y)=d(x,y)\}$ the segment with ends $x,y$. A set is metrically convex if it contains $[x,y]$ f…

Let $F_t$ be a one-parameter iterated function system satisfying properties (H1), (H2), and (H3) of Section 4. If $A_{\bullet}=A^{\bullet}$ for some upper transition attractor of…

Dimension conjecture. For all $0 < \lambda < \mu < 1$ with $\lambda + \mu > 1$, the set $A_{\lambda,\mu}$ has dimension strictly greater than $1$.

Top-boundary conjecture. For all $0 < \lambda < \mu < 1$ with $\lambda + \mu > 1$, the set $\partial_{top}(A_{\lambda,\mu})$ is the graph of a continuous, strictly increasing funct…

Attractor-generation conjecture. Given a hyperbolic IFS with probabilities, a sequence $\sigma$ is an element of $S$—that is, $\sigma$ generates the attractor of the IFS as describ…

Let $\Phi=\{\varphi_j(x)=\lambda_jx+t_j\}_{j=0}^m$ be a self-similar iterated function system on the real line, let $p=(p_j)_{j=0}^m$ be a probability vector, and let $\mu$ be its…

Let $\mathcal{M}$ be the connectivity locus and let $\mathcal{M}_0$ be the Thurston set, both contained in the unit disk $\mathbb{D}$. Let $U$ be the connected component of…

Let $\mathcal{M}$ and $\mathcal{M}_0$ be the connectivity locus and Thurston set, respectively, for the iterated function system generated by $\mathfrak{s}_-$ and $\mathfrak{s}_+$.…

Let $F_t$ be a bounded one-parameter affine family with attractors $A_t$, and let $t_0$ be the threshold at which attractors cease to exist. A compact set $A^*$ is an upper transit…

Let $F\subset[0,\infty)$ be the dimension spectrum of a CIFS, and let $\dim_x(F)$ denote its local Hausdorff dimension at $x\in F$. Three-type classification conjecture. Exactly on…

Local-dimension regularity conjecture. The map

Let $K\subset[0,\infty)$, and write its mirror image as $\sup(K)-K=\{\sup(K)-x:x\in K\}$. A dimension spectrum is a set arising as the set of Hausdorff dimensions of the limit sets…

Let $\Phi$ be an iterated function system and let $(p_i)_{i\in I}$ be a probability vector. For a finite word $\mathbf{a}=(a_1,\ldots,a_n)\in I^n$, write…

Fourier completeness criterion. The Fourier frequencies $\{e_n\}_{n\in\mathbb{N}_0^d}$ are total in $L^2(\mu)$ if and only if $v$ is non-constant, namely, not proportional to…

Let $U\subset\mathbb R^k$ be the parameter space of the iterated function systems $\Phi_u$, and let $\mathfrak m$ be a slowly decaying $\sigma$-invariant ergodic probability measur…

Let $U\subset\mathbb R^k$ be an open parameter space, let $\Phi_u$ be the parameterised iterated function systems described above, and let $X_u$ be their attractors. For a slowly d…

Let $U$ be an open subset of $\mathbb{R}^k$, and for each $u\in U$ let $\Phi_u=\{\phi_{i,u}(x)=A_i(u)x+t_i(u)\}_{i=1}^l$ be an iterated function system with attractor $X_u$. Let…